Approximation van der Waals

The inaccuracy of the theory for long chains and low densities can be attributed to the van der Waals approximation in the theory. For a bulk fluid, the excess internal energy, Uex, per molecule is given by... 

Figure 2.3 Schematic T-dependent shifts in Z for a C02-like gas (i.e., Van der Waals approximation), showing the gradual shifts toward ideality at all pressures.

Figure 2.5 Compressibility factor Z(P) for C02 at 40°C (cf. Fig. 2.2), comparing the Van der Waals approximation (solid line) with experimental values (circles, dotted line) and with the ideal gas approximation (dashed line).

It was shown by J. C. Maxwell that a horizontal line can be drawn through the Van der Waals loop region in such a way that the area enclosed above the line in the upward loop exactly matches that enclosed below the line in the downward loop ( Maxwell s equal-area construction ). As shown in Fig. 2.10b, this horizontal line (say, at pressure P0) can be taken as the Van der Waals approximation to the actual condensation plateau, bounded on the left by the steeply sloping liquid branch, and on the right by the more gently sloping gaseous branch of the isotherm. The three points where this horizontal line P = P0 crosses the Van der Waals isotherm may be obtained as the roots of the cubic polynomial P = P(V) for P = P0, i.e., as solutions of the equation... 

Van der Waals himself originally conjectured that universal functional relationships of the form (2.56) remain valid beyond the Van der Waals approximation. This conjecture, the principle (often law ) of corresponding states, can be stated as follows ... 

The van der Waals approximation is built on the idea that is a background molecular field that scarcely fluctuates during the course of the thermal motion of the system composed of the distinguished solute and solution. In that situation, is so tightly concentrated that the only significant parameter is... 

We will later consider the approximation that affects the transition from Eq. (4.4) to Eq. (4.6) in detail. But this result would often be referred to as first-order perturbation theory for the effects of - see Section 5.3, p. 105 - and we will sometimes refer to this result as the van der Waals approximation. The additivity of the two contributions of Eq. (4.1) is consistent with this form, in view of the thermodynamic relation pdpi = dp (constant T). It may be worthwhile to reconsider Exercise 3.5, p. 39. The nominal temperature independence of the last term of Eq. (4.6), is also suggestive. Notice, however, that the last term of Eq. (4.6), as an approximate correction to will depend on temperature in the general case. This temperature dependence arises generally because the averaging ((... ))i. will imply some temperature dependence. Note also that the density of the solution medium is the actual physical density associated with full interactions between all particles with the exception of the sole distinguished molecule. That solution density will typically depend on temperature at fixed pressure and composition. 

The first two terms here constitute the van der Waals approximation as discussed above. The succeeding term is a correction that lowers this free energy. The thermodynamic excess chemical potential is then obtained by averaging the Boltzmann factor of this conditional result using the isolated solute distribution function sj 0l ). 

This is a primitive van der Waals approximation for the indicated conditional expectation. The associated physical argument is that, with the density specified, further assessment of fluctuations is less important. 

The van der Waals approximation discussed in Section LA applies the mean field approximation in the solid phase in the same way as in the fluid phase. Baus and co-workers [150,151] have recently presented an alternative formulation in which the localization of the molecules in the solid phase is taken into account. They have applied this to the understanding of trends in the phase diagrams of systems of hard spheres with attractive tails as the range of attractions is changed. For the mean field term in the solid phase they use the static lattice energy for the given interaction potential and crystal lattice. A similar approach was used earlier to incorporate quad-rupole-quadrupole interactions into a van der Waals theory calculation of the phase diagram of carbon dioxide [152]. 

Fig. 4.6 The Joule-Thomson coefficient for H2 in the Van der Waals approximation (Equation (4.26)) at a pressure ofp = 0.1 MPa (solid line) and p=10MPa (dashed line).

The next three theories, listed in Table 3, all make use of the so-called van der Waals approximation to relate the interaction parameters for the mixture to the parameters associated with the individual like and unlike interactions. They differ mainly in their choice of equation of state. The theory of Leland, Rowlinson, and Sather used the experimental equation of state of methane whereas the two other theories use explicit analytic equations of state, those of van der Waals and Guggenheim respectively. The LRS theory appears to... 

Simple expressions for the flexoelectric coefficients in the nematic phase can be obtained using the generalized van der Waals approximation. In this approximation the intermolecular attraction is taken into account in the mean-field approximation while the steric repulsion is accounted for by taking into consideration short-range steric correlations via the excluded... 

Figure 7.1 Interfacial density profile AplAp( zb 00) as a function of z/. —, the mean-field (van der Waals) approximation given by eq 7.21 . , calculated in renormalization-group theory and given by eq 7.70.

This value is independent of p and is used with (3.13) to give the van der Waals approximation to the surface tension,... 

A second line of improvement of the original van der Waals approximation, which can also he tested on the penetrable-sphere model. is the substitution of a two-density for a one-density theory, as set out in H 3.3 and as applied to the two-component lattice gas in S 5.4. 

Fio. 5.10. The surface tension for a one-dimensional potential (.v - 1) as calculated, (1) from the mean-field approximation, (2) from the original van der Waals approximation, (3) from the modification of Yang et al., and (4) from the two-density" theory. 

We showed in 4.4-4.S that the functional expansion of the free energy in terms of the direct correlation fnnetion led naturally to a justification of the van der Waals approximation of (Thapter 3, but with the coefficient of the square of the density gradient, m(p), now a function... 

We conclude this chapter by comparing the surface tension of a Lennard-Jones liquid, as calculated from the theories above, with that found by computer simulation. Figure 7.4 shows the computer results of Fig. 6.4 as a best single line. The four theoretical curves are one solution of the YBG equation, one approximation based on the direct correlation function, one modified van der Waals approximation, and one... 

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